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Modal basis approaches in shape and topology optimization of frequency response problems
Grégoire Allaire, Georgios Michailidis
To cite this version:
Grégoire Allaire, Georgios Michailidis. Modal basis approaches in shape and topology optimization of frequency response problems. International Journal for Numerical Methods in Engineering, Wiley, 2018, 113 (8), pp.1258-1299. �hal-01354162�
Modal basis approaches in shape and topology optimization of frequency response problems
Gr´egoire Allaire
CMAP, Ecole Polytechnique, CNRS, Universit´e Paris-Saclay, 91128 Palaiseau, France
([email protected]) Georgios Michailidis
SIMaP-Universit´e de Grenoble, INPG, 38000 Grenoble, France
([email protected]) July 14, 2016
Abstract
The optimal design of mechanical structures subject to periodic excitations within a large frequency interval is quite challenging. In order to avoid bad performances for non-discretized frequencies, it is necessary to finely discretize the frequency interval, leading to a very large number of state equations. Then, if a standard adjoint-based approach is used for optimization, the computational cost (both in terms of CPU and memory storage) may be prohibitive for large problems, especially in three space dimensions. The goal of the present work is to introduce two new non-adjoint approaches for dealing with frequency response problems in shape and topology optimization. In both cases, we rely on a classical modal basis approach to compute the states, solutions of the direct problems. In the first method, we do not use any adjoint but rather directly compute the shape derivatives of the eigenmodes in the modal basis. In the second method, we compute the adjoints of the standard approach by using again the modal basis. The numerical cost of these two new strategies are much smaller than the usual ones if the number of modes in the modal basis is much smaller than the number of discretized excitation frequencies. We present numerical examples for the minimization of the dynamic compliance in two and three space dimensions.
Keywords: shape and topology optimization, frequency response, level-set method
1 Introduction
Shape and topology optimization techniques [1, 5, 9] find today extensive applications in industry related to product design. Its incorporation in the design cycle can significantly reduce the required time for the conception of a mechanical part and improve its performance. Especially when the mechanical framework of the application is complicated enough, relying only on the experience and intuition of engineers frequently proves to be not efficient enough. In such cases, performance- driven automated design techniques can provide a powerful remedy, allowing designers to enrich their knowledge and result in better designs.
Frequency response problems appear in a wide variety of industries, such as the automotive and the aeronautic sectors. Such problems are characterized by a Helmholtz state equation, featuring
the excitation frequency and possibly some damping, and furthermore by the fact that usually one is not interested by a single frequency but rather by a whole interval of excitation frequencies.
Discretization of the frequency range yields a large number of Helmholtz state equations. Contrary to multiple loads optimization where the state equation is the same and only the right hand sides vary, frequency response optimization leads to a collection of different partial differential equations or rigidity matrices. As a consequence, the computational cost of frequency response optimization is very large and even prohibitive for industrial applications with large-scale problems.
This is all the more an issue since the frequency interval must be finely discretized otherwise intermediate frequencies in a gap between two discretized frequencies may lead to completely suboptimal performances.
In order to reduce the computational cost of frequency response optimization, a classical ap- proach is to work with a modal basis. In other words, for a given structure, one computes its first eigenfrequencies and eigenmodes (without damping) in a range covering the entire excitation frequency interval. Then, the Helmholtz state equations (with damping) are solved by a Galerkin method with this modal basis. In the context of shape and topology optimization, this additionally requires the solution of an adjoint problem for every eigenfrequency. Since the number of eigen- frequencies of the modal basis can be quite large, the total cost of this approach can be indeed prohibitive for industrial applications.
There is already a vast literature for topology optimization of frequency response problems and we give a brief and non exhaustive account of it. Early works were done in the framework of the homogenization method: Ma et al. [19], [20] minimized the mean compliance in a frequency interval using a direct and a modal analysis, while Min et al. [21] used an optimality criteria method to minimize the dynamic compliance, working in the time domain. In the context of plate thickness optimization, Jog [16] worked on the dynamic compliance and the frequency amplitude.
Later, many more works appeared using the SIMP method. Tcherniak [32] presented a method, based on modal analysis, for the design of resonating structures. Olhoff et al. [23] and Yoon [34]
studied the dynamic compliance minimization. A very interesting paper was written by Jensen [15] in order to accelerate the sensitivity computations: he proposed to use Pad`e approximants for the solution of dynamic problems, avoiding the costly calculation of adjoint states for many excitation frequencies. Finally, in the level-set framework (that we adopt in the present article) the first work about frequency response problems is that of Shu et al. [29].
Here, our main goal is to present two new methods for the treatment of frequency response problems, which do not require the computation of adjoint states, and thus accelerate significantly the optimization process and make feasible the application of shape and topology optimization on industrial frequency response problems. Although the proposed methods can be applied in the framework of density-based methods too, we use here the level-set method for the shape description [2, 3, 25, 28, 33], in order to benefit both from its geometric advantage of a clear definition of a shape and to avoid possible ghost modes in the modal analysis, localized in region of intermediate densities.
The content of the present paper is as follows. Section 2 is devoted to a general presentation of the mechanical framework and of the optimization setting for frequency response problems.
Section 3 is a brief introduction to the basic ingredients for shape and topology optimization via the level-set method. For reasons of completeness, the standard adjoint approach is recalled in Section 4. The main conclusion is that it leads to as many adjoint states as there are discrete excitation frequencies, which is totally prohibitive from a numerical point of view. Section 5 describes what is more or less the state of the art in frequency response problems. A modal basis approach is used to compute the solutions of the Helmholtz state equations. By considering the spectral equations as the effective state equations, the optimization process leads to consider new adjoint equations, the number of which is roughly the number of eigenfrequencies in the modal basis. Although it is a serious improvement with respect to the previous standard approach of
Section 4, it is still a very costly strategy in terms of computation time and memory storage. Our first new self-adjoint method is presented in Section 6: it is based on the approximation of the eigenmodes’ shape derivatives in the modal basis and requires no adjoint equations. Note that this method, as well as the next one, can cope with damping as long as the damping operator is diagonal in the modal basis, as is often the case in numerical practice. A second new self-adjoint method is proposed in Section 7. Its principle is quite simple: it starts from the standard shape derivative obtained in Section 4 and replace the direct and adjoint states by their approximation in the modal basis, therefore eliminating the need of solving any adjoint equations. A comparison of all these approaches is performed in Section 8 in terms of complexity or operation counts. It shows that our two proposed self-adjoint methods outperform the other approaches as soon as the number of discretized excitation frequencies and the number of eigenfrequencies in the modal basis are large. Of course, a key issue for numerical efficiency is to have access to a fast and accurate algorithm to compute the modal basis. Section 9 is an assessment of the modal basis approach.
We discuss convergence issues in terms of the number of eigenmodes and of the discretization step for the excitation frequency interval. Finally, numerical results in two and three space dimensions are shown in Section 10, which validates our proposed approach. We focus on dynamic compliance minimization problems but other objective functions would work as well in our setting. We also check that multiple eigenvalues are not an issue from a numerical point of view, although it is known to be a delicate point when it comes to their differentiability properties.
2 Setting of the problem
2.1 State equation
Consider a structure, occupying a bounded domain Ω⊂Rd, withd= 2,3, which vibrates under the application of a periodic time-harmonic loadf(x, t) at some part of its boundary ΓN ⊂∂Ω.
The structure is fixed at ΓD ⊂∂Ω, while the rest of its boundary, denoted Γ, is free and subject to optimization. The corresponding displacement fieldu(x, t) is a solution of the system
ρ(x)¨u+c(x) ˙u−div (A e(u)) = 0 in Ω×R+, u = 0 on ΓD×R+, A e(u)
n = f(t) on ΓN ×R+, A e(u)
n = 0 on Γ×R+,
(1)
whereρ(x)>0 is the material density,c(x)≥0 the damping function,Athe isotropic Hooke’s law, e(u) = (∇u+ (∇u)T)/2 the strain tensor and the dot ( ˙ ) denotes derivation with respect to time.
Since we are looking for a time-harmonic displacementu(x, t), we did not include initial conditions in (1). To facilitate the analysis, we work with complex-valued functions (withi=√
−1). Assume the complex loading to be of the type
f(x, t) =F(x)eiωt, F =fRe+ifIm, fRe, fIm∈R, (2) whereω >0 is the excitation frequency andF the complex loading amplitude. Similarly we look for a complex time-harmonic solution of (1) which reads as
u(x, t) =U(x)eiωt, U =uRe+iuIm, uRe, uIm∈R. (3)
One recovers a real-valued solution of (1) by taking the real part Re(u) =uRecos(ωt)−uImsin(ωt).
Substituting (3) in (1), we obtain the equation satisfied by the displacement amplitudeU:
−ω2ρU+iωcU−div (A e(U)) = 0 in Ω, U = 0 on ΓD, A e(U)
n = F on ΓN, A e(U)
n = 0 on Γ.
(4)
In the sequel we shall always work in the frequency domain, rather than in the time domain.
Namely, we use (4) as the state equation, instead of (1). Separating the real and imaginary parts, we get the following two coupled systems of PDE’s:
−ω2ρuRe−ωcuIm−div (A e(uRe)) = 0 in Ω, uRe = 0 on ΓD, A e(uRe)
n = fRe on ΓN, A e(uRe)
n = 0 on Γ,
(5)
and
−ω2ρuIm+ωcuRe−div (A e(uIm)) = 0 in Ω, uIm = 0 on ΓD, A e(uIm)
n = fIm on ΓN, A e(uIm)
n = 0 on Γ.
(6)
Remark 2.1. For given loadsfRe, fIm ∈L2(ΓN)d and in the presence of damping, i.e. c(x)≥0 and c 6= 0, the system of equations (5), (6) has a unique solution uRe, uIm ∈ H1(Ω)d (see e.g.
Lemma 2.6.6 in [22]). In the undamped case (c= 0), a unique solution exists when the frequency of the external loading does not coincide with an eigenfrequency of the structure. Note that in the sequel, we shall often replace the damping multiplicative coefficient c(x) by a diagonal operator C in the modal basis (see Section 5 for more details).
Remark 2.2. For simplicity, and because it is enough in many applications, we consider only surface loads onΓN. However, it is not a restriction and all our analysis in the present paper can be extendedmutatis mutandisto the case of bulk loads in a non-optimizable part of the domain Ω (the case of loads touching the optimizable free boundary is slightly different, although amenable to our approach).
Remark 2.3. After discretization, for example using the finite element method, equations (5) and (6) become
K−ω2M −ωC ωC K−ω2M
uRe
uIm
= fRe
fIm
, (7)
which, with obvious notations, can also be written in the form
S(ω)U =F with S(ω) =K+iωC−ω2M, (8)
where the complex matrixS(ω)is called the dynamic stiffness matrix. The matrices K, C, M are real symmetric andK, M are positive definite. Very often, at the discrete level, the matrix C is replaced by a linear combination ofK andM.
2.2 Optimization problem
Our goal is to find a shape Ω, belonging to an admissible set Uad that minimizes an objective functionJ(Ω), which depends on the shape through the solution (uRe, uIm) of equation (4). The shape optimization problem reads
Ω∈Uminad
J(Ω). (9)
Very often, we shall abuse notations and write J(Ω) =J
uRe(Ω), uIm(Ω) .
A famous example of such a functional, for frequency response applications, is the so-calleddy- namic compliance, which reads
J(Ω) = Z ωmax
ωmin
Z
ΓN
(fIm·uRe(ω)−fRe·uIm(ω))ds dω, (10) where [ωmin, ωmax] denotes the interval of excitation frequencies, 0< ωmin< ωmax. More general objective functions can be considered as well: our entire approach in the sequel can easily be extended. Their distinctive feature is that they depend on a full range of frequencies and not just a single one. In numerical practice, this frequency range will be discretized and theω-integral in (10) will be replaced by a sum over finitely many excitation frequencies ˜ωi, 1 ≤ i ≤ Nω. The number of discrete frequencies Nω is usually very large (typically of the order of 100 to 1000, corresponding to a frequency step of one Hertz), which makes the evaluation of the objective function quite expensive since it requires to solveNωstate equations of the type (4).
Remark 2.4. For the sake of completeness, we give here a brief physical interpretation of the dynamic compliance for a specific excitation frequencyω. Defining the instantaneous input power due to the loading as
Pinp= Z
ΓN
( ˙u·f)ds= Z
ΓN
−ω(uResin(ωt) +uImcos(ωt))·(fRecos(ωt)−fImsin(ωt))ds, the energy change over one excitation cycle due to the loading reads
∆Einp= Z 2πω
0
Pinpdt=π Z
ΓN
(uRe·fIm−uIm·fRe)ds.
Omitting the factorπ, the dynamic compliance is thus defined as
Cdyn= Z
ΓN
(uRe·fIm−uIm·fRe)ds.
3 Shape and topology optimization framework
In general, a shape and topology optimization method is characterized by two major ingredients: a method to describe the shape and a method to update it, optimizing its performance with respect to some pre-defined criteria. In this work, following the lead of [3, 33] we use the level-set method for the shape description and the Hadamard method of shape differentiation in order to deduce a descent direction, briefly described in the rest of this section.
3.1 Level-set method
The level-set method, developped by Osher and Sethian [26], uses an implicit representation of an evolving front as the zero level-set of an auxiliary functionφ. More precisely, assuming that the domain Ω of interest is a subset of a large working domainD, the level-set representation of Ω can be defined as
φ(x) = 0 ⇔x∈∂Ω∩D, φ(x)<0 ⇔x∈Ω, φ(x)>0 ⇔x∈ D\Ω
.
(11)
The advection of the front (or shape boundary) with a normal velocityV(x, t) is described in the level-set framework by the well-known Hamilton-Jacobi transport equation:
∂φ
∂t +V(x, t)|∇φ|= 0, (12)
using an explicit second order upwind scheme [24, 27]. In our optimization setting the timet∈R+ can be interpreted, after discretization, as a descent step.
3.2 Shape sensitivity approach
In shape and topology optimization, the normal velocityV(x, t) used for the shape evolution in (12) is chosen so that the objective function decreases during the (pseudo-)time evolution. In a gradient flow approach, or gradient-based optimization method, one first needs to compute a shape derivative of the objective function by using the classical Hadamard method [1], [14], [30], [31].
Starting from a smooth reference open set Ω, we consider variations of the type Ωθ= Id+θ
Ω, withθ∈W1,∞(Rd,Rd).
Definition 3.1. The shape derivative ofJ(Ω)atΩis defined as the Fr´echet derivative inW1,∞(Rd,Rd) at 0 of the applicationθ→J (Id+θ)Ω
, i.e.
J (Id+θ)Ω
=J(Ω) +J0(Ω)(θ) +o(θ) with lim
θ→0
|o(θ)|
kθk = 0, whereθ→J0(Ω)(θ)is a continuous linear form onW1,∞(Rd,Rd).
A classical result (Hadamard’s structure theorem) states that the shape derivative J0(Ω)(θ) depends only on the normal trace θ·n on the boundary ∂Ω. In fact, for a great variety of functionals, the shape derivative can be written in the form
J0(Ω)(θ) = Z
∂Ω
θ(s)·n(s)V(s)ds, (13)
where the integrandV depends on the specific objective function and boundary conditions. Then, a descent direction can be found by advecting the shape in the directionθ(s) =−tV(s)n(s) for a small enough descent stept >0. For the new shape Ωt= ( Id +tθ) Ω, if V 6= 0, we can formally write
J(Ωt) =J(Ω)−t Z
∂Ω
V(s)2ds+O(t2)< J(Ω), which guarantees a descent direction for small positivet.
Assumption. In the sequel, we assume that only the free boundary Γ is allowed to be optimized and that the Dirichlet and Neumann parts of the boundary ΓD and ΓN are kept fixed. In other words, we assume that all vector fieldsθ satisfyθ= 0 on ΓD and ΓN.
4 Standard adjoint approach
In this Section, we obtain the shape derivative of the dynamic compliance (10) by following the standard adjoint approach. More precisely, we rely on the Lagragian method of C´ea [8] for calcu- lating a shape derivative. We recall that we work in the frequency domain. The results obtained here will serve as a basis for the second non-adjoint method we propose in Section 7.
To simplify the presentation, we compute the shape derivative of the dynamic compliance for one single excitation frequencyω, i.e.
Jω(Ω) = Z
ΓN
(fIm·uRe(ω)−fRe·uIm(ω))ds . The shape derivative of (10) is then derived as
J0(Ω)(θ) = Z ωmax
ωmin
Jω0(Ω)(θ)dω.
Proposition 4.1. Assume that the damping is not zero, namely c(x)≥0 andc 6= 0. The shape derivative of the single-frequency dynamic compliance is
Jω0(Ω)(θ) = Z
Γ
−ω2ρ(uRe·pRe+uIm·pIm)−ωc(uIm·pRe−uRe·pIm) +Ae(uRe)·e(pRe) +Ae(uIm)·e(pIm)
θ·n ds,
(14)
where(pRe, pIm)is the adjoint state, solution of
−ω2ρpRe+ωcpIm−div (A e(pRe)) = 0 inΩ, pRe = 0 on ΓD, A e(pRe)
n = −fIm on ΓN, A e(pRe)
n = 0 on Γ,
(15)
and
−ω2ρpIm−ωcpRe−div (A e(pIm)) = 0 in Ω, pIm = 0 on ΓD, A e(pIm)
n = fRe on ΓN, A e(pIm)
n = 0 on Γ.
(16)
Remark 4.1. The numerical computation of the dynamic compliance (10) and its shape derivative (14) requires to compute the state(uRe, uIm)and the adjoint(pRe, pIm)for every discrete frequency ωin the frequency range[ωmin, ωmax]. This is too costly for most real-life industrial problems. For this reason, modal basis approaches are usually prefered, as we shall explain in the next section.
Proof. Since the Dirichlet boundary ΓD is fixed, we can introduce a Sobolev spaceV, defined by V =
v∈H1(Rd)d such thatv= 0 on ΓD , (17) which is independent of the choice of the shape Ω. Following the method of C´ea, we define a Lagrangian which is the sum of the objective function and of the variational formulation of (4),
L(Ω, v, q) = Jω(vRe, vIm) +
Z
Ω
−ω2ρvRe·qRe−ωcvIm·qRe+A e(vRe)·e(qRe) dx−
Z
ΓN
fRe·qReds
+ Z
Ω
−ω2ρvIm·qIm+ωcvRe·qIm+A e(vIm)·e(qIm) dx−
Z
ΓN
fIm·qImds,
wherevRe, vIm∈V plays the role of the state,qRe, qIm∈V are the adjoints or Lagrange multipliers for the state equation and, for the dynamic compliance,
Jω(vRe, vIm) = Z
ΓN
(fIm·vRe−fRe·vIm)ds.
We fix a domain Ω and consider the optimality conditions for the LagrangianL at the optimal point (Ω, v∗, q∗). Obviously, the conditions
h ∂L
∂qRe(Ω, v∗, q∗), φi= 0, h ∂L
∂qIm(Ω, v∗, q∗), φi= 0,
for a smooth test functionφ∈V, reveal that (vRe∗ , vIm∗ ) = (uRe, uIm) is the unique solution of the variational formulation for the coupled system of equations (5) and (6).
The partial derivative ofLwith respect tovRe, in the direction of a test functionφ∈V, at the optimal point, reads
h ∂L
∂vRe(Ω, v∗, q∗), φi= h∂Jω
∂vRe(vRe∗ , v∗Im), φi+ Z
Ω
−ω2ρφ·qRe∗ +ωcφ·q∗Im+A e(φ)·e(qRe∗ ) dx
= Z
ΓN
fIm·φ ds+ Z
Ω
−ω2ρφ·qRe∗ +ωcφ·q∗Im+A e(φ)·e(qRe∗ ) dx.
(18) A similar formula can be obtained for the partial derivative ofLwith respect tovIm. Setting (18) equal to zero (as well as the other formula), we get that (qRe∗ , qIm∗ ) = (pRe, pIm) is the solution of the adjoint system (15) and (16).
Finally, the shape derivative of the objective function is just the shape partial derivative of the Lagrangian at the optimal point (v∗, q∗) [1], [8] (this requires that the solutionv∗ of the coupled system of equations (5) and (6) is shape differentiable, which is a classical result). In other words, the shape derivative ofJω reads
Jω0(Ω)(θ) =L0(Ω, v∗, q∗)(θ) = Z
Γ
(θ·n)( −ω2ρuRe·q∗Re−ωcuIm·q∗Re+Ae(uRe)·e(qRe∗ )
−ω2ρuIm·q∗Im+ωcuRe·qIm∗ +Ae(uIm)·e(q∗Im))ds, which is nothing but (14).
Remark 4.2. Proposition 4.1 still holds true when there is no damping,c= 0, provided thatω is not a resonance frequency (corresponding to an eigenvalue of the problem). In such an undamped case, the problem is self-adjoint since(pRe, pIm) = (−uIm, uRe). In general, forc6= 0, the optimiza- tion of the dynamic compliance is not a self-adjoint problem, meaning that the adjoint(pRe, pIm)is not a simple combination of the state(uRe, uIm). There are however a few cases where it is indeed self-adjoint. In case fIm = 0, by comparison we find pRe =uIm and pIm =uRe. Therefore the shape derivative (14) reads
Jω0(Ω)(θ) = Z
Γ
−2ω2ρuRe·uIm−ωc(|uIm|2− |uRe|2) + 2Ae(uRe)·e(uIm)
θ·n ds. (19) In casefRe = 0, by comparison we find pRe =−uIm, pIm =−uRe and the shape derivative (14) reads
Jω0(Ω)(θ) = Z
Γ
2ω2ρuRe·uIm+ωc(|uIm|2− |uRe|2)−2Ae(uRe)·e(uIm)
θ·n ds. (20) Remark 4.3. In the spirit of Remark 2.3, in matrix notation, the adjoint equation can be written
K−ω2M −ωC ωC K−ω2M
pIm pRe
= fRe
−fIm
, (21)
or, equivalently,S(ω)P =iF with the notation P =pRe+ipIm and the dynamic stiffness matrix S(ω) =K+iωC−ω2M. The matrix in (21) is the same as in (7) for the direct problem so its factorization can be kept in order to minimize the overhead of solving for an adjoint. However, there are as many linear systems to solve than frequenciesω in the discretization of the dynamic compliance.
5 Modal analysis using an adjoint method
The modal decomposition allows to solve large-scale dynamics problems in reasonable time. In frequency response problems, the great majority of publications and commercial softwares use a modal analysis, coupled with an adjoint state for every mode considered. For reasons of complete- ness, although classical in the literature, we present here the detailed shape derivation using this approach.
5.1 Modal decomposition
We introduce the modal basis for the elasticity problem (4) without damping, i.e. c = 0. The eigenvalues are the squares of the eigenfrequenciesωj >0,j≥1, labelled by increasing order with repeated multiplicities. The eigenmodesrj are real vector-valued functions which satisfy
−div (A e(rj))−ωj2ρrj = 0 in Ω, rj = 0 on ΓD, A e(rj)
n = 0 on ΓN ∪Γ,
(22) and are normalized by
Z
Ω
ρrj·rjdx= 1. (23)
The damped elasticity equation (4) can not be diagonalized by these eigenmodes in full generality because the damping term c(x) is not a spectral combination of the inertia term ρ(x) and of the elasticity operator div(A e(·)). However, in engineering practice, the damping term is often assumed to be diagonalizable. We adhere to this setting and define a damping linear operatorC which is somehow a linear combination of the inertia term and of the elasticity operator and will be defined more precisely later (see (26) below) by its spectral decomposition. In other words, we replace (4) by
−ω2ρU+iωC(U)−div (A e(U)) = 0 in Ω, U = 0 on ΓD, A e(U)
n = F on ΓN, A e(U)
n = 0 on Γ.
(24)
The complex amplitudeU is decomposed in the basis formed by the eigenvectorsrj, such that U(x) =
∞
X
j=1
ajrj(x) with aj =ajRe+iajIm and ajRe, ajIm∈R. (25) To compute the coordinatesaj, we substitute (25) in (24) and take its variational formulation with the test functionrk (thek-th eigenfunction). It leads to
∞
X
j=1
aj
−ω2 Z
Ω
ρrj·rkdx+iω Z
Ω
C(rj)·rkdx+ Z
Ω
Ae(rj)·e(rk)dx
= Z
ΓN
fRe·rkds+i Z
ΓN
fIm·rkds.
By using the orthogonality property of the eigenfunctions, we have, forj6=k, Z
Ω
ρrj·rkdx= 0 and Z
Ω
Ae(rj)·e(rk)dx= 0, and forj =k
Z
Ω
ρrk·rkdx=Mkk>0 and Z
Ω
Ae(rk)·e(rk)dx=Kkk >0.
We now make precise our definition of the assumed damping operatorC which is diagonal in the same basis with
Z
Ω
C(rj)·rkdx=
0 ifj6=k,
Ckk= 2Mkkωkξk ifj=k, (26) whereξk >0 denotes the damping ratio for the eigenmodek. We thus deduce
ak =< fRe, rk >ΓN +i < fIm, rk>ΓN
(Kkk−ω2Mkk) +iωCkk
,
with the following notation
< f, rk>ΓN= Z
ΓN
f·rkdx .
Now, our normalization assumption implies thatMkk= 1 andKkk=ω2k, which yields ak= < fRe, rk>ΓN +i < fIm, rk >ΓN
(ω2k−ω2) +i(2ωωkξk)
= (< fRe, rk>ΓN +i < fIm, rk >ΓN) (ωk2−ω2)−i(2ωωkξk) (ωk2−ω2)2+ (2ωωkξk)2 .
(27)
Note thatak is always finite because its denominator cannot vanish, even ifω=ωk, sinceξk 6= 0 by assumption. Eventually, the complex amplitude functionU = uRe+iuIm is obtained as an explicit function of the full set of eigenfrequencies and eigenmodes{ωj, rj}∞j=1
uRe=
∞
X
j=1
< fRe, rj>ΓN (ωj2−ω2)+< fIm, rj >ΓN 2ωωjξj (ωj2−ω2)2+ (2ωωjξj)2
!
rj (28)
and
uIm=
∞
X
j=1
< fIm, rj >ΓN (ωj2−ω2)−< fRe, rj >ΓN 2ωωjξj
(ωj2−ω2)2+ (2ωωjξj)2
!
rj. (29)
Remark 5.1. In the above analysis, the damping coefficients have been modelled by (26), i.e., Ckk = 2Mkkωkξk. Although this is a practical and popular assumption in engineering practice [10], there are different models which could equally be considered. For example, another classical model amounts to assuming that the damping matrix is proportional to a linear combination of the mass and stiffness matrices, i.e. C = αM +βK, with coefficients α, β > 0, independent of the eigenfrequenciesωk.
Remark 5.2. Of course, in numerical practice the series in formulas (28) and (29) are truncated to a finite number of modes j ≤nmod in order to obtain a computable approximation of U. A discussion of the numerical cost is given later in Section 8.
5.2 Shape derivative
We now give a different formula for the shape derivative of the dynamic compliance, based on the modal decomposition of the previous subsection. The main idea is to replace the objective functionJ(Ω), which depends on the solution U of the damped wave equation (4), by the same objective functionJmod(Ω) which is written in terms of the modal decomposition (28) and (29) ofU and thus is a function of the eigenvalues and eigenmodes. To simplify the presentation, we again consider the single-frequency dynamic compliance
Jω(Ω) = Z
ΓN
(fIm·uRe(ω)−fRe·uIm(ω))ds .
Proposition 5.1. Assume that the damping is not zero and is modelled by (26), withξj >0for all modes. Assume that all eigenfrequenciesωj are simple. The shape derivative of the single-frequency dynamic compliance is
Jω0(Ω)(θ) = Z
Γ
θ·n
∞
X
j=1
Ae(rj)·e(qj)−ωj2ρrj·qj+µjρ|rj|2
ds . (30)
whereqj is the adjoint state, solution of the adjoint equation (33), andµj is a Lagrange multiplier defined by (38).
Remark 5.3. We emphasize that Proposition 5.1 is valid only if the eigenfrequencies ωj and eigenmodes rj are shape differentiable. This is usually achieved by assuming that all eigenvalues are simple (multiplicity equal to one). Therefore, the analysis presented here is valid only if multiple eigenvalues are not present.
Remark 5.4. Formulas (14) and (30) for the shape derivative of the dynamic compliance should be equivalent although it is not easy to check. Of course, in the context of Proposition 5.1, we made a different spectral assumption on the damping than in Proposition 4.1, so this comparison has to be made only when there is no dampingc= 0.
Remark 5.5. The shape derivative of the dynamic compliance (10) is recovered from (30) by integrating with respect toω, namely
J0(Ω)(θ) = Z ωmax
ωmin
Jω0(Ω)(θ)dω= Z
Γ
θ·n
∞
X
j=1
Ae(rj)·e(qej)−ωj2ρrj·qej+fµjρ|rj|2 ds ,
where
qej= Z ωmax
ωmin
qjdω and fµj = Z ωmax
ωmin
µjdω .
Of course, the adjointsqj depend on the excitation frequencyω. However, by inspecting the adjoint equation (33), the excitation frequency appears only in its right hand side. In other words, for different excitation frequenciesω, the adjointsqj(ω)share the same differential operator or rigidity matrix. Therefore, the averaged adjointqej can be computed as the solution of the adjoint equation (33), where the right hand side is also averaged with respect to ω, and only one adjoint equation per eigenfrequency has to be solved (whatever the number of excitation frequencies). In numerical practice, only a finite numbernmodof eigenmodes is used to approximate the shape derivative (30).
Therefore, from a computation point of view, formula (30) is superior to the previous formula (14) since it requires less adjoints because the number of eigenmodes nmod is usually much smaller than the number of discretized excitation frequencies in the definition of the dynamic compliance.
Neverteless, the overall CPU cost of the method of Proposition 5.1 is still quite expensive. See Section 8 for more details.
Proof. Any objective functionJ(Ω) =J(uRe(Ω), uIm(Ω)), which is defined in terms of the solution (uRe, uIm) of (5), (6), can also be considered as a function of the full set of eigenfrequencies and eigenmodes{ωj, rj}∞j=1 by virtue of the modal decomposition (28) and (29). We denote byJmod this function, defined by
J(uRe, uIm) =Jmod
{ωj, rj}∞j=1 .
To compute the shape derivative ofJω(Ω), we use once more the method of C´ea but applied to the functionJωmod, considering the spectral equation (22) as a constraint, instead of the elasticity